$A$ charged particle (electron or proton) is introduced at the origin $(x=0, y=0, z=0)$ with a given initial velocity $\overrightarrow{v}$. $A$ uniform electric field $\overrightarrow{E}$ and magnetic field $\vec{B}$ are given in columns $I, II$ and $III$, respectively. The quantities $E_0, B_0$ are positive in magnitude.
Column $I$ Column $II$ Column $III$ $(I)$ Electron with $\overrightarrow{v}=2 \frac{E_0}{B_0} \hat{x}$ $(i)$ $\overrightarrow{E}=E_0 \hat{z}$ $(P)$ $\overrightarrow{B}=-B_0 \hat{x}$ $(II)$ Electron with $\overrightarrow{v}=\frac{E_0}{B_0} \hat{y}$ $(ii)$ $\overrightarrow{E}=-E_0 \hat{y}$ $(Q)$ $\overrightarrow{B}=B_0 \hat{x}$ $(III)$ Proton with $\overrightarrow{v}=0$ $(iii)$ $\overrightarrow{E}=-E_0 \hat{x}$ $(R)$ $\overrightarrow{B}=B_0 \hat{y}$ $(IV)$ Proton with $\overrightarrow{v}=2 \frac{E_0}{B_0} \hat{x}$ $(iv)$ $\overrightarrow{E}=E_0 \hat{x}$ $(S)$ $\overrightarrow{B}=B_0 \hat{z}$
$(1)$ In which case will the particle move in a straight line with constant velocity?
$(2)$ In which case will the particle describe a helical path with axis along the positive $z$ direction?
$(3)$ In which case would the particle move in a straight line along the negative direction of $y$-axis (i.e., move along $-\hat{y}$)?
| Column $I$ | Column $II$ | Column $III$ |
| $(I)$ Electron with $\overrightarrow{v}=2 \frac{E_0}{B_0} \hat{x}$ | $(i)$ $\overrightarrow{E}=E_0 \hat{z}$ | $(P)$ $\overrightarrow{B}=-B_0 \hat{x}$ |
| $(II)$ Electron with $\overrightarrow{v}=\frac{E_0}{B_0} \hat{y}$ | $(ii)$ $\overrightarrow{E}=-E_0 \hat{y}$ | $(Q)$ $\overrightarrow{B}=B_0 \hat{x}$ |
| $(III)$ Proton with $\overrightarrow{v}=0$ | $(iii)$ $\overrightarrow{E}=-E_0 \hat{x}$ | $(R)$ $\overrightarrow{B}=B_0 \hat{y}$ |
| $(IV)$ Proton with $\overrightarrow{v}=2 \frac{E_0}{B_0} \hat{x}$ | $(iv)$ $\overrightarrow{E}=E_0 \hat{x}$ | $(S)$ $\overrightarrow{B}=B_0 \hat{z}$ |
$(1)$ In which case will the particle move in a straight line with constant velocity?
$(2)$ In which case will the particle describe a helical path with axis along the positive $z$ direction?
$(3)$ In which case would the particle move in a straight line along the negative direction of $y$-axis (i.e., move along $-\hat{y}$)?
- A
- B
- C
- D
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